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<li><a href="../packages/JAMA%0D%0ACholesky%20decomposition%20class%0D%0AFor%20a%20symmetric,%20positive%20definite%20matrix%20A,%20the%20Cholesky%20decomposition%0D%0Ais%20an%20lower%20triangular%20matrix%20L%20so%20that%20A%20=%20L*L'.html"><i class="icon-folder-open"></i> JAMA
Cholesky decomposition class
For a symmetric, positive definite matrix A, the Cholesky decomposition
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Class to obtain eigenvalues and eigenvectors of a real matrix</a></li>
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<li><a href="../packages/JAMA%0D%0AFor%20an%20m-by-n%20matrix%20A%20with%20m%20&gt;=%20n,%20the%20LU%20decomposition%20is%20an%20m-by-n%0D%0Aunit%20lower%20triangular%20matrix%20L,%20an%20n-by-n%20upper%20triangular%20matrix%20U,%0D%0Aand%20a%20permutation%20vector%20piv%20of%20length%20m%20so%20that%20A(piv,:)%20=%20L*U.html"><i class="icon-folder-open"></i> JAMA
For an m-by-n matrix A with m &gt;= n, the LU decomposition is an m-by-n
unit lower triangular matrix L, an n-by-n upper triangular matrix U,
and a permutation vector piv of length m so that A(piv,:) = L*U</a></li>
<li><a href="../packages/JAMA%0D%0AFor%20an%20m-by-n%20matrix%20A%20with%20m%20&gt;=%20n,%20the%20QR%20decomposition%20is%20an%20m-by-n%0D%0Aorthogonal%20matrix%20Q%20and%20an%20n-by-n%20upper%20triangular%20matrix%20R%20so%20that%0D%0AA%20=%20Q*R.html"><i class="icon-folder-open"></i> JAMA
For an m-by-n matrix A with m &gt;= n, the QR decomposition is an m-by-n
orthogonal matrix Q and an n-by-n upper triangular matrix R so that
A = Q*R</a></li>
<li><a href="../packages/JAMA%0D%0AFor%20an%20m-by-n%20matrix%20A%20with%20m%20&gt;=%20n,%20the%20singular%20value%20decomposition%20is%0D%0Aan%20m-by-n%20orthogonal%20matrix%20U,%20an%20n-by-n%20diagonal%20matrix%20S,%20and%0D%0Aan%20n-by-n%20orthogonal%20matrix%20V%20so%20that%20A%20=%20U*S*V'.html"><i class="icon-folder-open"></i> JAMA
For an m-by-n matrix A with m &gt;= n, the singular value decomposition is
an m-by-n orthogonal matrix U, an n-by-n diagonal matrix S, and
an n-by-n orthogonal matrix V so that A = U*S*V'</a></li>
<li><a href="../packages/JAMA%0D%0APythagorean%20Theorem:%0D%0Aa%20=%203%0D%0Ab%20=%204%0D%0Ar%20=%20sqrt(square(a)%20+%20square(b))%0D%0Ar%20=%205%0D%0Ar%20=%20sqrt(a%5E2%20+%20b%5E2)%20without%20under.overflow.html"><i class="icon-folder-open"></i> JAMA
Pythagorean Theorem:
a = 3
b = 4
r = sqrt(square(a) + square(b))
r = 5
r = sqrt(a^2 + b^2) without under/overflow</a></li>
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<i class="icon-custom icon-method"></i> Methods
                    <ul>
<li class="method public "><a href="#method___construct" title="__construct :: Construct the singular value decomposition"><span class="description">Construct the singular value decomposition</span><pre>__construct()</pre></a></li>
<li class="method public "><a href="#method_cond" title="cond :: Two norm condition number"><span class="description">Two norm condition number</span><pre>cond()</pre></a></li>
<li class="method public "><a href="#method_getS" title="getS :: Return the diagonal matrix of singular values"><span class="description">Return the diagonal matrix of singular values</span><pre>getS()</pre></a></li>
<li class="method public "><a href="#method_getSingularValues" title="getSingularValues :: Return the one-dimensional array of singular values"><span class="description">Return the one-dimensional array of singular values</span><pre>getSingularValues()</pre></a></li>
<li class="method public "><a href="#method_getU" title="getU :: Return the left singular vectors"><span class="description">Return the left singular vectors</span><pre>getU()</pre></a></li>
<li class="method public "><a href="#method_getV" title="getV :: Return the right singular vectors"><span class="description">Return the right singular vectors</span><pre>getV()</pre></a></li>
<li class="method public "><a href="#method_norm2" title="norm2 :: Two norm"><span class="description">Two norm</span><pre>norm2()</pre></a></li>
<li class="method public "><a href="#method_rank" title="rank :: Effective numerical matrix rank"><span class="description">Effective numerical matrix rank</span><pre>rank()</pre></a></li>
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<i class="icon-custom icon-property"></i> Properties
                    <ul></ul>
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<li class="nav-header private">» Private
                    <ul>
<li class="property private "><a href="#property_U" title="$U :: Internal storage of U."><span class="description"></span><pre>$U</pre></a></li>
<li class="property private "><a href="#property_V" title="$V :: Internal storage of V."><span class="description"></span><pre>$V</pre></a></li>
<li class="property private "><a href="#property_m" title="$m :: Row dimension."><span class="description"></span><pre>$m</pre></a></li>
<li class="property private "><a href="#property_n" title="$n :: Column dimension."><span class="description"></span><pre>$n</pre></a></li>
<li class="property private "><a href="#property_s" title="$s :: Internal storage of singular values."><span class="description"></span><pre>$s</pre></a></li>
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<span class="divider">\</span><a href="../classes/SingularValueDecomposition.html">SingularValueDecomposition</a>
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<th>package</th>
<td><a href="../packages/JAMA%0D%0AFor%20an%20m-by-n%20matrix%20A%20with%20m%20&gt;=%20n,%20the%20singular%20value%20decomposition%20is%0D%0Aan%20m-by-n%20orthogonal%20matrix%20U,%20an%20n-by-n%20diagonal%20matrix%20S,%20and%0D%0Aan%20n-by-n%20orthogonal%20matrix%20V%20so%20that%20A%20=%20U*S*V'.%0D%0AThe%20singular%20values,%20sigma%5B%24k%5D%20=%20S%5B%24k%5D%5B%24k%5D,%20are%20ordered%20so%20that%0D%0Asigma%5B0%5D%20&gt;=%20sigma%5B1%5D%20&gt;=%20...%20&gt;=%20sigma%5Bn-1%5D.%0D%0AThe%20singular%20value%20decompostion%20always%20exists,%20so%20the%20constructor%20will%0D%0Anever%20fail.%20%20The%20matrix%20condition%20number%20and%20the%20effective%20numerical%0D%0Arank%20can%20be%20computed%20from%20this%20decomposition..html">JAMA
For an m-by-n matrix A with m &gt;= n, the singular value decomposition is
an m-by-n orthogonal matrix U, an n-by-n diagonal matrix S, and
an n-by-n orthogonal matrix V so that A = U*S*V'.
The singular values, sigma[$k] = S[$k][$k], are ordered so that
sigma[0] &gt;= sigma[1] &gt;= ... &gt;= sigma[n-1].
The singular value decompostion always exists, so the constructor will
never fail.  The matrix condition number and the effective numerical
rank can be computed from this decomposition.</a></td>
</tr>
<tr>
<th>author</th>
<td><a href="">Paul Meagher</a></td>
</tr>
<tr>
<th>license</th>
<td><a href="">PHP v3.0</a></td>
</tr>
<tr>
<th>version</th>
<td>1.1</td>
</tr>
</table>
<h3>
<i class="icon-custom icon-method"></i> Methods</h3>
<a id="method___construct"></a><div class="element clickable method public method___construct" data-toggle="collapse" data-target=".method___construct .collapse">
<h2>Construct the singular value decomposition</h2>
<pre>__construct($Arg) : \Structure</pre>
<div class="labels"></div>
<div class="row collapse"><div class="detail-description">
<div class="long_description"><p>Derived from LINPACK code.</p></div>
<h3>Parameters</h3>
<div class="subelement argument"><h4>$Arg</h4></div>
<h3>Returns</h3>
<div class="subelement response">
<code>\Structure</code>to access U, S and V.</div>
</div></div>
</div>
<a id="method_cond"></a><div class="element clickable method public method_cond" data-toggle="collapse" data-target=".method_cond .collapse">
<h2>Two norm condition number</h2>
<pre>cond() : \max(S)/min(S)</pre>
<div class="labels"></div>
<div class="row collapse"><div class="detail-description">
<div class="long_description"></div>
<table class="table table-bordered"><tr>
<th>access</th>
<td>public</td>
</tr></table>
<h3>Returns</h3>
<div class="subelement response"><code>\max(S)/min(S)</code></div>
</div></div>
</div>
<a id="method_getS"></a><div class="element clickable method public method_getS" data-toggle="collapse" data-target=".method_getS .collapse">
<h2>Return the diagonal matrix of singular values</h2>
<pre>getS() : \S</pre>
<div class="labels"></div>
<div class="row collapse"><div class="detail-description">
<div class="long_description"></div>
<table class="table table-bordered"><tr>
<th>access</th>
<td>public</td>
</tr></table>
<h3>Returns</h3>
<div class="subelement response"><code>\S</code></div>
</div></div>
</div>
<a id="method_getSingularValues"></a><div class="element clickable method public method_getSingularValues" data-toggle="collapse" data-target=".method_getSingularValues .collapse">
<h2>Return the one-dimensional array of singular values</h2>
<pre>getSingularValues() : \diagonal</pre>
<div class="labels"></div>
<div class="row collapse"><div class="detail-description">
<div class="long_description"></div>
<table class="table table-bordered"><tr>
<th>access</th>
<td>public</td>
</tr></table>
<h3>Returns</h3>
<div class="subelement response">
<code>\diagonal</code>of S.</div>
</div></div>
</div>
<a id="method_getU"></a><div class="element clickable method public method_getU" data-toggle="collapse" data-target=".method_getU .collapse">
<h2>Return the left singular vectors</h2>
<pre>getU() : \U</pre>
<div class="labels"></div>
<div class="row collapse"><div class="detail-description">
<div class="long_description"></div>
<table class="table table-bordered"><tr>
<th>access</th>
<td>public</td>
</tr></table>
<h3>Returns</h3>
<div class="subelement response"><code>\U</code></div>
</div></div>
</div>
<a id="method_getV"></a><div class="element clickable method public method_getV" data-toggle="collapse" data-target=".method_getV .collapse">
<h2>Return the right singular vectors</h2>
<pre>getV() : \V</pre>
<div class="labels"></div>
<div class="row collapse"><div class="detail-description">
<div class="long_description"></div>
<table class="table table-bordered"><tr>
<th>access</th>
<td>public</td>
</tr></table>
<h3>Returns</h3>
<div class="subelement response"><code>\V</code></div>
</div></div>
</div>
<a id="method_norm2"></a><div class="element clickable method public method_norm2" data-toggle="collapse" data-target=".method_norm2 .collapse">
<h2>Two norm</h2>
<pre>norm2() : \max(S)</pre>
<div class="labels"></div>
<div class="row collapse"><div class="detail-description">
<div class="long_description"></div>
<table class="table table-bordered"><tr>
<th>access</th>
<td>public</td>
</tr></table>
<h3>Returns</h3>
<div class="subelement response"><code>\max(S)</code></div>
</div></div>
</div>
<a id="method_rank"></a><div class="element clickable method public method_rank" data-toggle="collapse" data-target=".method_rank .collapse">
<h2>Effective numerical matrix rank</h2>
<pre>rank() : \Number</pre>
<div class="labels"></div>
<div class="row collapse"><div class="detail-description">
<div class="long_description"></div>
<table class="table table-bordered"><tr>
<th>access</th>
<td>public</td>
</tr></table>
<h3>Returns</h3>
<div class="subelement response">
<code>\Number</code>of nonnegligible singular values.</div>
</div></div>
</div>
<h3>
<i class="icon-custom icon-property"></i> Properties</h3>
<a id="property_U"> </a><div class="element clickable property private property_U" data-toggle="collapse" data-target=".property_U .collapse">
<h2></h2>
<pre>$U : array</pre>
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</div>
<a id="property_V"> </a><div class="element clickable property private property_V" data-toggle="collapse" data-target=".property_V .collapse">
<h2></h2>
<pre>$V : array</pre>
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</div>
<a id="property_m"> </a><div class="element clickable property private property_m" data-toggle="collapse" data-target=".property_m .collapse">
<h2></h2>
<pre>$m : int</pre>
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</div>
<a id="property_n"> </a><div class="element clickable property private property_n" data-toggle="collapse" data-target=".property_n .collapse">
<h2></h2>
<pre>$n : int</pre>
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</div>
<a id="property_s"> </a><div class="element clickable property private property_s" data-toggle="collapse" data-target=".property_s .collapse">
<h2></h2>
<pre>$s : array</pre>
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